In closing of the second part, we will explain how to apply the theory we have developed so far to the example \(\mathrm {U}_q(\infty )\)  from quantum unitary groups \(\mathrm {U}_q(n)\) . As an application of the notion of quantized character functions, we will give a clearer interpretation of Gorin's \(q\) -Schur generating functions (see [1]}) in terms of \(\mathrm {U}_q(\infty )\) . We try to make it clear that all the computations can essentially be done by means of quantized universal enveloping algebras \(U_q\mathfrak {gl}(n)\)  (although the formulation of spherical unitary representations certainly needs the operator algebraic notion of compact quantum groups due to its character of unitary representations).
